We know for sure there is an 11 year sunspot cycle, and there is no a priori reason why there many not be other tiny regular and irregular variations superimposed on it, leading to a minor cyclic or random variation in solar output. We know of many different types of variable star, although to astronomically detect variability at stellar distances requires intrinsic variation which would be catastrophic to life if it were exhibited by our sun.

However, to detect these alleged variations would first require we collect precise solar data over long (centuries) periods of time, and that we were able to model out and correct for purely geometrical effects such as minute changes in the ellipticity and eccentricity of earth's orbit, rotation of the line of apsides, obliquity of the ecliptic, and changes in the amount and periodicity of precession and nutation; all effects which are chaotic in nature and due to solar system objects perturbing each other. These effects no doubt interact in complex ways to modulate solar insolation, which then is converted to climate variations by the extremely complex heat engine of the hydro-, litho-, atmo-, bio-spheres. In short, we just don't know. One thing is for sure, it is highly unlikely the solar system affects the sun gravitationally. The sun's family has been described as the sun, Jupiter, and assorted debris; and Jupiter's sidereal period is 11.86 years.

What we do know is that earth's climate has varied enormously over geologic time, although how much of this variation is due to an energy input signal and how much is the result of how that signal is damped or amplified by the earth itself, is anyone's guess.